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<div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>









<a NAME="E1:36"/><i><font color="Green">E30</font></i>: 
( ( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font></span>)</span> implies  ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font> ) ) &amp; (  ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font> ) implies <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font></span>)</span> ) &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> holds <br/> ( <font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font> iff  ex <font color="Olive">h</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">E</font>,<font color="Olive">h</font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">H</font> ) ) ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:36"/>
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">H</font> implies  ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font> st <br/>( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) ) &amp; <font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">H</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E31</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:36_1"/><i><font color="Green">E32</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">H</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E6:36_1"/>
for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E2:36_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E3:36_1_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:36_1"/>
<font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><a NAME="E3:36"/><i><font color="Green">E35</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) ) &amp; <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">H</font> )
 ;<br/>

<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/><a NAME="E2:36_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E3:36_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:36_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:36"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">y</font>,<font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:36"/>
for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div>
